[Journal of Astrophysics and Astronomy] Multimessenger Astrophysics (Volume 39, Issue 4, August 2018)

103 Pages·2018·18.651 MB·English

by  Debades Bandyopadhyay, Pratik Majumdar (eds.)

J.Astrophys.Astr.(2018)39:44 ©IndianAcademyofSciences
DOI10.1007/s12036-018-9541-6
Editorial
Recent discoveries in astrophysics and cosmology the first Indian observatory in space and gravitational
have generated tremendous excitement in the scien- observatory LIGO, India which is in the making.
tific world. The first high-energy neutrinos of cosmic Furthermore, India is participating and taking lead
originwererecordedbytheIceCube,agiganticcubic- in building the Square Kilometre Array (SKA), the
kilometre-sizeddetectorintheAntarcticiceattheSouth next generation radio telescope to be co-located in
Pole.Theprizecatchofgravitationalwaves(GW)from South Africa and Australia, Thirty Metre optical Tele-
mergingblackholesinLIGOdetectorsforthefirsttime scope (TMT) and has strong co-operation with the
in2015opensanewwindowtotheUniverse.Thedis- members of the next generation imaging atmospheric
coveryofgravitationalwavesfromthemergingbinary Cerenkov detector consortium, Cerenkov Telescope
neutron star in August last year followed by its detec- Array (CTA) with generous support from the Depart-
tion across the electromagnetic (EM) spectrum is an ment of Atomic Energy and Department of Science
outstandingeventinthehistoryofmankind.Thedetec- and Technology. Saha Institute of Nuclear Physics
tionoflightfromfirststarsjust180millionyearsafter (SINP) has been involved in PICASSO/PICO Dark
theBigBang,hailedasanothergreatbreakthrough,was Matter search experiment in SNOLab, Canada since
reportedrecentlybyagroupofradioastronomers.This 2009 as well as study of very high energy gamma-
observation also hinted at the presence of dark matter. ray sources observed with the Major Atmospheric
Alltheseobservationsmarkaneweraofmultimessen- GammaImagingCerenkov(MAGIC)telescopeexper-
gerastronomy:theexplorationoftheUniversethrough imentsince2015.Furthermore,SINPalongwithother
combininginformationfromamultitudeofcosmicmes- institutions is spearheading the Dark Matter search
sengers:electromagneticradiation,gravitationalwaves, experiment in the underground laboratory to be set up
neutrinosandcosmicrays. inJaduguda.
The questions that can be explored through Multimessenger astronomy hence requires a co-
multimessenger observations concern the dynamics of ordination of a global network of multimessenger
exploding stars, the formation and evolution of black instruments, to develop multimessenger observational
holes,theoriginofcosmicrays,relativisticjets,super- strategies and data analysis and an interdisciplinary
massiveblackholesintheheartsofgalaxies,colliding efforttointerpretobservationsandconstraintheoretical
black holes and neutron stars and many others. Mul- models. All this requires tight collaborations between
timessenger astronomy also allows us to address the the different GW/EM and neutrino/cosmic-rays com-
questionofwhywearehereinthefirstplace,byshed- munities. The Astroparticle and Cosmology Division
ding light on the origin of heavy elements and the ofSINPthusaimstobringtogetherthesecommunities
evolutionofgalaxiesandtheUniverse. together in order to start a discussion on how to co-
In the next decade, multimessenger astronomy will ordinatethesediverseastrophysicalactivitiesunderthe
probe the rich physics of transient phenomena in the umbrella of multimessenger astrophysics. This is our
sky, such as the merger of neutron stars and/or black tribute to Professor Meghnad Saha on his 125th birth
holes, gamma-ray bursts, active galactic nuclei and anniversary.
core-collapse supernovae. India is making a steady
progress in multimessenger astronomy with the Giant GuestEditors
Metre-Wave Radio Telescope (GMRT) near Pune, a DebadesBandyopadhyay
marvelcreatedbyProfessorGovindSwarup;AstroSat, PratikMajumdar
Publishedonline11August2018
J.Astrophys.Astr.(2018)39:40 ©IndianAcademyofSciences
https://doi.org/10.1007/s12036-018-9533-6
Calculation of the transport coefficients of the nuclear pasta phase
RANANANDI1,∗ andSTEFANSCHRAMM2
1TataInstituteofFundamentalResearch,Mumbai400005,India.
2FrankfurtInstituteforAdvancedStudies,60438FrankfurtamMain,Germany.
∗
Correspondingauthor.E-mail:[email protected]
MSreceived31May2018;accepted11June2018;publishedonline11July2018
Abstract. Wecalculatethetransportcoefficientsoflow-densitynuclearmatter,especiallythenuclearpasta
phase,usingquantummoleculardynamicssimulations.Theshearviscosityaswellasthethermalandelectrical
conductivities are determined by calculating the static structure factor of protons for all relevant density,
temperatureandprotonfractions,usingsimulationdata.Itisfoundthatallthetransportcoefficientshavesimilar
ordersofmagnitudeasfoundearlierwithoutconsideringthepastaphase.Ourresultsarethusincontrasttothe
commonbeliefthatthepastalayerishighlyresistiveandthereforehaveimportantastrophysicalconsequences.
Keywords. Neutronstars—transportproperties—moleculardynamics.
1. Introduction supernova explosion (Horowitz et al. 2004). On the
other hand, the electron-pasta scattering is supposed
Based on the composition a neutron star (NS) can be to greatly influence the transport properties like shear
divided into several parts. At the surface there is a viscosityandthermalandelectricalcoductivitiesofNS
thinenvelopecontainingmostlyHandHeionsandFe crustandtherefore,canplayacrucialroleinunderstand-
atoms. The outer crust starts at ∼104 gcm−3 when ing the phenomena of cooling (Horowitz et al. 2015),
atoms get fully ionized and form a Coulomb lattice magneticfielddecay(Ponsetal.2013),crustaloscilla-
embedded in an electron gas. With increasing density tions(Chugunov&Yakovlev2005),etc.ofNS.
the nucleus become increasingly neutron-rich due to Thepastaphasewasinitiallystudiedbystaticmeth-
electron capture process. At ∼4 × 1011 gcm−3, the ods using few specific shapes (Ravenhall et al. 1983;
nucleus become so neutron rich that neutrons begin to Lorenz et al. 1993; Oyamatsu 1993; Newton & Stone
drip out of the nuclei (Baym et al. 1971; Rüster et al. 2009).However,sincetheshapesarenotknownapri-
2006;Nandi&Bandyopadhyay2011).Thismarksthe ori it is important to adopt dynamical approach that
end of the outer crust and the beginning of the inner allowsarbitrary nuclear shapes. After the first work of
crust.Intheinnercrustnucleiareimmersedinbothan Maruyamaetal.(1998),severalauthorshavestudiedthe
electron gas and a neutron gas (Haensel 2001; Nandi characteristics of pasta phase using molecular dynam-
et al. 2011). With further increase in density, nuclei icssimulationofdifferenttypes(Horowitzetal.2004;
comecloserandat∼1014 gcm−3 theymergetogether Schneider et al. 2013; Dorso et al. 2012; Watanabe
to form uniform matter of neutrons, protons, electrons et al. 2003; Nandi & Schramm 2016, 2017; Schramm
andmuons.Justbeforethetransitiontouniformmatter, & Nandi 2017a,b). In this article, we study the trans-
nucleitakevariouscomplicatedshapesduetothecom- portpropertiesofthenuclearpastaphaseusingquantum
petition between Coulomb energy and surface energy. moleculardynamics(QMD)simulation.
These exotic shapes are collectively known as nuclear
‘pastaphase’(Ravenhalletal.1983).
The study of pasta phase is crucial to understand 2. Formalism
various astrophysical phenomena. For example, the
scattering of neutrinos from the pasta phase in core- InQMD,thewavefunctionofanucleonisrepresented
collapsesupervovaecansignificantlyaffecttheneutrino by a Gaussian wave packet with time-independent
transport, that plays a critical role in the eventual width.Theinteractionbetweennucleonsisdescribedby
40 Page 2 of 5 J.Astrophys.Astr.(2018)39:40
a Skyrme-like Hamiltonian (Nandi & Schramm 2018; 90
T=1
Chikazumietal.2001) 80 T=2
T=3
H=T+VPauli+VSkyrme+Vsym+VMD+VSurface+VCoul, 70 Yp=0.5 TT==45
(1) 60 ρ=0.1ρ0
whereT isthekineticenergy,VPauli isthephenomeno- (q)p 50
S 40
logical Pauli potential, V is the Skyrme-like
Skyrme
interaction between nucleons, V denotes the 30
MD
momentum-dependent part, V is the isospin- 20
sym
dependentpart,VSurface isthepotentialthatdependson 10
the density gradient and VCoul is the Coulomb interac- 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
tion.The explicitexpressionsfor all the potentialscan
q (fm-1)
befoundinNandiandSchramm(2018).
Figure 1. Staticstructurefactorversusmomentumtransfer
The dominant contribution to transport properties
forprotonsatρ =0.1ρ ,Y =0.5anddifferenttemperatures
likeshearviscosity(η)andthermalandelectricalcon- 0 p
(T =1–5)MeV.
ductivities (κ, σ) of pasta phase comes from electron-
ionscatteringaselectronsarethemostimportantcarri-
350
ersofchargeandmomentuminthiscondition.Allthese T=1
T=2
transportcoefficientscaneasilybecalculatedwhenthe 300 T=3
T=4
bsteattwicesetnrupcrtoutroenfsaicstokrno(Swpn(.qW))ethcaatlcduelsactreibSes(qc)orfrreolmatitohne 250 ρY=p0=.04.ρ30 T=5
p
autocorrelationfunction(Nandi&Schramm2018) q) 200
(p
S (q) = 1 (cid:4)ρ (q,t)∗ρ (q,t)(cid:5), (2) S 150
p p p
N
p 100
where ρ (q,t) is the Fourier transform of proton
p
50
number density, which is calculated using positions of
the nucleons at time t. The average is taken over sim- 0
0.1 0.2 0.3 0.4 0.5 0.6
ulationtimeaswellasallthedirectionsofmomentum q (fm-1)
transfer q = 2π(l, m, n), where l, m, n are inte-
L Figure 2. Same as in Fig. 1, but for Y = 0.3 and ρ =
p
gers and L is the length of the cubic simulation box 0.4ρ .
0
determined from the number of nucleons used in the
simulationandthedensityas L = (N/ρ)1/3.Toincor-
poratelong-range(smallq)correlationsamongprotons form factor and the screening effects of other ions.
oneneedstohaveenoughnumberofparticles.Wetake At ρ = 0.1ρ0, nucleons form spherical clusters. With
4096 nucleons at ρ = 0.1ρ (ρ = 0.168 fm−3 is the increasing T more and more nucleons evaporate from
0 0
nuclearsaturationdensity),andincreaseitwithdensity clusters making the system increasingly uniform and
keeping L fixedat62.47fm.Therefore,forthehighest henceleadtosmallerSp(qpeak).Aninterestingbehavior
density (0.6ρ0) considered here we have 24,576 parti- of Sp(q)isseeninthepastaphaseregion(ρ (cid:2) 0.2ρ0).
cles.Tokeepthesimulationruntimewithinreasonable In Fig. 2, we plot Sp(q) for asymmetric nuclear mat-
limit,weperformsimulationusingGPUplatform. ter (Yp = 0.3) at ρ = 0.4ρ0. From the figure, we
see that S (q) increases very sharply at T = 2 MeV.
p
Thisisbecausecylindricalstructuresfoundatthisden-
3. Results sity at low T merge together to form almost perfect
equidistant slabs at T = 2, as can be seen from the
We calculate S (q) and subsequently transport snapshot shown in Fig. 3. With further increase in T,
p
coefficientsforvariousdensity,temperaturesandproton these slabs gradually merge to form bubble phase and
fraction (Y ) relevant for various astrophysical scenar- S (q) decreases, as a consequence. In Fig. 4, we plot
p p
ios. In Fig. 1, we plot S (q) as a function of q for S (q)foralldensityandtemperaturesconsideredhere.
p p
symmetric nuclear matter at ρ = 0.1ρ and tempera- A careful observation shows that at low-density S (q)
0 p
turesT = 1−5MeV.TheS (q =q )isproportional decreases with temperature for all three values of Y .
p peak p
tothesizeoftheclustersbutgetcorrectedbythenuclear However in the pasta phase region (ρ (cid:2) 0.2ρ ), the
0
J.Astrophys.Astr.(2018)39:40 Page 3 of 5 40
Figure 3. Simulationsnapshotforneutron(green)andproton(red)distributionsatρ =0.4ρ ,Y =0.3andT =2MeV.
0 p
Y =0.5 Y =0.3 Y =0.1
p p p
100
)
q
(
p
S
10
T=0
T=1
T=2
T=3
T=4
T=5
1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6
ρ/ρ ρ/ρ ρ/ρ
0 0 0
Figure 4. S (q)asafunctionofdensityforY andT.
p p
S (q) rises at intermediate temperatures as discussed 4. Conclusion
p
beforeinconnectiontoFig.2.
Using the calculated structure factors we finally We have calculated the transport coefficients namely
calculatethetransportcoefficients.InFig.5andFig.6, shearviscosityandthermalandelectricalconductivities
we display the results of shear viscosity and thermal of low-density nuclear matter at various astrophysical
conductivity for asymmetric nuclear matter with Y = conditions, using quantum molecular dynamics simu-
p
0.3 and Y = 0.1, which are typical values for super- lations. Under these conditions the electrons are most
p
nova matter and neutron star inner crust, respectively. importantcarriersandtheirtransportislimitedmainly
It is observed that although there are irregularities in byelectron–ionscattering,whichwecalculatebydeter-
the pasta phase region both the transport coefficients miningthestaticstructureforprotons(S (q))usingour
p
generally increase with density and temperature. Most simulationdata.AlthoughS (q)showssomeirregular-
p
importantly,the order of magnitude of both the coeffi- ities in the pasta region, all the transport coefficients
cients are same as found earlier without considering arefoundtohavesimilarordersofmagnitudeasfound
the pasta phase (Flowers & Itoh 1976; Nandkumar withoutconsideringthepastaphase.Ourresultsthere-
& Pethick 1983). Similar behavior is also found for fore contradict the speculation that the pasta layer
electricalconductivity. is highly resistive and bear important astrophysical
40 Page 4 of 5 J.Astrophys.Astr.(2018)39:40
14.5 23
T=0
T=1
T=2
T=3
T=4
14
T=5 22.5
Y =0.3
p
cm/s) 13.5 m/s/K) 22
η (g/0 13 κ (g/c
1 0
og g1 21.5
l o
l
12.5
21
12
20.5
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6
ρ/ρ ρ/ρ
0 0
Figure 5. ShearviscosityandthermalconductivityofnuclearmatterasafunctionofdensityandtemperatureatY =0.3.
p
15.5 22.8
T=0
T=1
T=2 22.6
15 T=3
T=4 22.4
T=5
14.5
22.2
Y =0.1
p
cm/s) 14 m/s/K) 212.28
η (g/0 13.5 κ (g/c 21.6
1 0
g 1
lo 13 og 21.4
l
21.2
12.5
21
12
20.8
11.5 20.6
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6
ρ/ρ ρ/ρ
0 0
Figure 6. Sameasinfigure5,butwithY =0.1.
p
consequences. However, longerand larger simulations Dorso C. O., Giménez Molinelli P. A., López J. A. 2012,
arerequiredtoconfirmthesefindings. Phys.Rev.C,86,055805
Flowers,E.,Itoh,N.1976,Astrophys.J.,206,218
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J.Astrophys.Astr.(2018)39:41 ©IndianAcademyofSciences
https://doi.org/10.1007/s12036-018-9532-7
Review
Entering the cosmic ray precision era
PASQUALEDARIOSERPICO
USMB,CNRS,LAPTh,Univ.GrenobleAlpes,74940Annecy,France.
E-mail:[email protected]
MSreceived5April2018;accepted18May2018;publishedonline11July2018
Abstract. HereweoutlinesomerecentactivitiesinthetheoryandphenomenologyofGalacticcosmicrays,
inthelightofthegreatprecisionofdirectcosmicraymeasurementsreachedinthelastdecade.Intheenergy
domainofinterest,rangingfromafewGeV/nucleontotensofTeV/nucleon,datahaverevealedsomenovel
featuresrequiringanexplanation.Weshallemphasizetheimportanceofamorerefinedmodeling,ofachieving
abetterassessmentoftheoreticaluncertaintiesassociatedtothemodels,andoftestingkeypredictionsspecific
ofdifferentmodelsagainsttherichdatasetsavailablenowadays.Despitethestillshakytheoreticalsituation,
several hints have accumulated suggesting the need to go beyond the approximation of a homogeneous and
non-dynamicaldiffusioncoefficientintheGalaxy.
Keywords. Cosmicrays—astroparticlephysics—interstellarmedium.
1. Introduction into a remarkable detail. For instance, the study of
non-lineareffectsandtheirimpactonsomeexpectations
Theevolutionofcosmicray(CR)astrophysicshasbeen ofdiffusiveshockaccelerationmodelsisnowamature
relatively slow, when compared with other branches sub-fieldoftheoreticalresearchofitsown.Butitisfair
of astronomy and astrophysics. This is not surprising, tosaythatnostringenttestofeitherthestandardormore
given the lack of positional information and the com- exotic models had been possible via charged CR mea-
plicatedpropagationthatmakethesourceidentification surements till recently, when a wealth of 21st century
andtheinterstellartransportcharacterizationsuchdiffi- experiments has significantly improved the precision
cultinversionproblems.Aboutadecadeago,afewmain of the observations, while extending their dynamical
questionsinCRphysicsandtheconsensualanswersto range.Takethefollowinglistofstatements:
themhadcrystallizedintoa‘standardframework’: • Weonlyhaveaccesstocosmicrayfluxes‘mod-
ulated’byheliosphere.
• How is CR acceleration taking place? Primarily • The positron flux is dominated by secondaries,
via‘diffusiveshockacceleration’. with propagation parameters (as opposed to
• In what type of objects? Predominantly (Galac- assumptionsonthesourceandmodelframework)
tic)supernovaremnants. constituting the dominant source of theoretical
• Wherearetheylocated?Whendidtheeventshap- uncertainty.
pen?RandomlyintheGalaxy,wellapproximated • Primarycosmicrayfluxeshavepower-lawspec-
byacontinuuminjectionterm,withasizemuch tra.
smallerthantypicalsource–Earthdistance. • Primaryspectrahaveuniversal(speciesindepen-
• How do CRs get to us, after leaving their dent)spectralindices.
acceleration sites? Diffusing into an externally
Thefirstitemhasbeendisprovenbytheuniqueexploit
assigned,roughlyscale-invariantturbulentmag-
of the Voyager Interstellar mission1 (see, Stone et al.
netizedinterstellarmedium(ISM).
2013, Cummings et al. 2016). The second item, usu-
Obviously,thisdoesnotmeanthatalternativescenarios allytakenforgrantedinmostphenomenologicalstudies
had not been occasionally considered. And, certainly,
some of the above-listed topics have been developed 1https://voyager.jpl.nasa.gov/mission/interstellar-mission/.
41 Page 2 of 8 J.Astrophys.Astr.(2018)39:41
over the past 30years, has not only been shaken by
newdata,notably—butnotexclusively—thecelebrated
‘PAMELApositronfractionrise’(Adrianietal.2009)
butappearsnowadaysverydoubtful(see,forinstance,
themini-reviewofSerpico2012).
The last two items have become unsteady nearly 7–
8yearsagobythemoreprecisemeasurementsavailable
(Ahnetal.2010,Yoonetal.2011,Adrianietal.2011),
with a trend still continuing today, notably thanks to
AMS-02(Aguilaretal.2015a,b).Alsointhelightofthe
importanceofsomeoftheseissuesforotherastroparti-
clephysicsapplications—notably,indirectdarkmatter
searches—a new scrutiny of the simplest theoretical
scenarios is ongoing. Ideally, theorists would like to
matchtheoreticaluncertaintieswithexperimentalones,
refining the level of predictions and improving our
understandingofthesehigh-energyphenomena.Atthe
same time, they face the challenge to come up with a
sufficientlypredictiveframework,notplaguedbyapro-
liferationofparameters,whichinthisfieldareoftentoo
hard or impossible to fix otherwise than by a fit to the
data.
2. Spectralbreaks
Inordertoillustratethistheoreticaltrend,wedescribe
Figure 1. The proton (top) and He (bottom) fluxes mea-
a specific example: the impact of the fact that primary
suredbyAMS-02(Aguilaretal.2015a,b)vs.rigidityR(ratio
cosmic ray fluxes in the GeV to TeV energy range,
ofmomentumtocharge,hencemeasuredinGigaVolts,GV),
in particular protons and He nuclei, do not manifest a rescaledbyR2.7,extractedviatheon-linecosmicraydatabase
simplepower-lawspectrum.Asalreadymentioned,the
tool http://lpsc.in2p3.fr/crdb/. It is visible to the naked eye
evidence in favour of spectral shapes closer to broken thattheslopeintensofGVto∼200GVaredifferentfromthe
power-lawshasaccumulatedoverthepastdecade,from slopesabovethelatterrigidityvalue.Alsonotetheremark-
the indications in balloon-borne experiments, such as ablysmallerrorbars.
ATIC(Panovetal.2009)andespeciallyCREAM(Yoon
et al. 2011), through the first measurements in a sin- ∂(cid:2)
τ (E): − K∇2(cid:2)= Q
gle space-based experiment, PAMELA (Adriani et al. diff ∂t
2011), till the recent high-precision determinations (cid:2)
⇒ = Q (atsteadystate).
by AMS-02 on board the International Space Station τ
diff
(Aguilaretal.2015a,b).InFig.1,thelatestproton(top (1)
panel)andhelium(bottompanel)fluxesfromtheAMS-
02 experiment are reported. They have been extracted If both the source term Q and the diffusion coeffi-
with the on-line cosmic ray database tool http://lpsc. cient K (with τdiff ∝ K−1) are power-laws in rigidity
in2p3.fr/crdb/(see,Maurinetal.2014),whichcanalso R, as customarily believed and theorized, then a puz-
be easily used to compare with the older datasets (not zle arises. This schematic exercise naturally suggests
reportedheretoavoidclutter). (classes of) solutions, where one drops one or several
Whatis‘wrong’withtheseobservations?Toassess, ofthefollowingassumptions(examplesofactualphys-
take the simplest expectation which, nonetheless, icalmotivationsforthatbelow,initalics):
matched data fairly well till recently. For stationary,
homogeneousandisotropicdiffusivepropagationprob- • Homogeneity (and possibly isotropy) of K.
lems,andobservationstakenatasinglelocation(i.e.the Example: multi-phase character of the Galactic
Earth) the diffusion operator ruling the flux (cid:2) can be interstellar medium (and nature of magneto-
effectivelyreplacedbya‘diffusiveconfinement’time hydro-dynamical(MHD)turbulence).
J.Astrophys.Astr.(2018)39:41 Page 3 of 8 41
• Power-law behaviour in K. Example: multiple 101 Stablelawα=4/3
sources/mechanismsfortheMHDturbulencein Stablelawα=5/3
theISM. 100 Gaussianlawσsim
• Power-law behaviour in Q. Example: multiple 10−1 Simulations
classes of sources or spectral feature of a single 10−2
sourceclass. 10−3
• Homogeneity in Q. Example: prominent local, 10−4
discretesources.
50
Before coming back to the first options in the final 0
−50
part of this article, we will briefly concentrate on the
0.0 0.2 0.4 0.6 0.8 1.0
latteroptiontoillustratesomerecenttheoreticalefforts log10(Ψ/Ψsim)
withintheabove-mentionedstrategy,whileaddressing Figure 2. ThebluehistogramisthepdfoftheGalacticCR
the reader to Serpico (2016) for a broader overview flux at 1TeV (vs.the flux normalized toits mean) obtained
of the alternatives, in particular, concerning multiple numerically via 106 Monte Carlo realizations of pure dif-
sourceorsourcespectraleffects. fusive transport. The dot-dashed blue line represents the
(highlyunsatisfactory)Gaussianapproximationfittedtothe
numerical results. The solid green line reports the theoret-
ical prediction based on fat-type stable law distribution for
3. Localsources
thelimitingcaseofaninfinitelythintwo-dimensionalmodel
oftheGalacticmagnetichalo,whereasthedashedredcurve
A number of publications have studied the possibility
corresponds to the 3D isotropic limit, expected to be valid
thattheCRspectralbreaksemergefroma‘local’source
respectivelyatlowandhighflux(dominatedrespectivelyby
contributionbecomingpredominantoveradiffusecon- farandnearsources).Thepercentresidualsbetweentheory
tributionrepresentativeofaGalacticaverage.Usually, andsimulationsaredisplayedinthepanelbelow,withbands
but not always, the local contribution is considered to showingtheir1-σPoissonerror.AdaptedfromGenolinietal.
dominateathigh-energy.Theemphasishasoftenbeen (2017a),whichweaddresstoforfurtherdetails.
on finding a viable fit, sometimes supplemented by a
qualitative assessment on the goodness of the model. TheCentralLimittheoremdoesnotapplyandthefamil-
Forinstance,onetypicallyneedsfastdiffusionandlow iarGaussianstatisticstoolboxcannotbeused.Wehave
supernova explosion rates for these scenarios to work, arguedthatageneralizedCentralLimittheoremholds,
whichhasoftenbeenarguedtobeintensionwithother and that the flux probability distribution functions are
observations. remarkably well approximated by ‘stable laws’, char-
One may however ask the more general question: acterized by analytically computable parameters. We
Howlikelyissuchahypothesisinitself,given‘Galactic testedtheseconclusionswithextensivenumericalsim-
variance’, i.e., the spatial discreteness (and impulsive ulations(seeFig.2foranexample).
time-dependence) of the sources? Conventional mod- Asaresult,wearrivedattwointerestingconclusions:
els,infact,replacetheactualsourceswithacontinuum
‘source jelly’, with a smoothly varying injection rate • Forcurrentlyviablehomogeneousandisotropic
perunitvolumeandtime.Thiscorrespondstoa‘coarse- diffusion models, the observed breaks only
grained’ensembleaverageoftheactualphysicalmodel, emergerarely,witharealisticupperlimitaround
andbyconstructiontheaveragetheoreticalexpectation 0.1%.Hence,suchexplanationsappeartorequire
matchesthepredictionobtainedinsuchasimplification. ahigh-degreeoffine-tuning.
But,assumingthatadiscrepancybetweenobservations • Even if this effect is probably insufficient to
and data is found, how safely can we attribute it to a account for the breaks, these ‘irreducible theo-
failure of the model? Couldn’t it be due to a relatively retical errors’ are not negligible anymore, given
large statistical fluctuation with respect to the average the precision of the data: by taking the exper-
prediction, which is in fact compatible with the model imental error σ reported by AMS-02 on the
exp
inamorerealisticcalculation? protonfluxmeasurement(Aguilaretal.2015a),
InGenolinietal.(2017a),wehaveoutlinedthefirst we estimate, for instance, that a 3 σ devia-
exp
elementsofsuchatheory.Thetaskismadenon-trivial tion from the average flux expectation at E ∼
by the fact that the theoretical probability distribution 50GeVisobtainedinabout5%ofthetheoretical
for the flux is of ‘fat-tail’ type, with infinite variance: realizations. Put otherwise, if the viability of a